Finite Group Databasebeta

← #6477801

Subgroups — #6477801

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35079C211yesmaximal, characteristic
2C2×#2347C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#33146C212yesmaximal, complemented
5#29973C212yesmaximal, complemented
6#29969C212yesmaximal, complemented
7#26015C214yesmaximal, complemented
8#26005C214yesmaximal, complemented
9#25997C214yesmaximal, complemented
10#25989C214yesmaximal, complemented
11#21944C211yesmaximal, characteristic
12#21940C211yesmaximal, characteristic
13#21932C211yesmaximal, characteristic
14#20702C211yesmaximal, characteristic
15#20699C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2857C2212yes—
20#2851C2212yes—
21#2667—21no—
22#2665—21no—
23#2569—21no—
24#2549—21no—
25#2442C2212yes—
26#2441C2212yes—
27#2440C2212yes—
28#2439C2212yes—
29#2437—22no—
30#2347—22no—
31#2347—21no—
32#2347—21no—
33#2347C2214yes—
34#2347C2212yes—
35#2347C2212yes—
36#2329—22no—
37C2×#389C2212yes—
38C2×#388—22no—
39C2×#383—21no—
40#2280C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4—21no—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#2101C2214yescomplemented
47#2100C2214yescomplemented
48#2088C2214yescomplemented
49#2087C2214yescomplemented
50#2087C2214yescomplemented
51#2086C2214yescomplemented
52#2059C2214yescomplemented
53#2057C2214yescomplemented
54#1989C2214yescomplemented
55#1988C2214yescomplemented
56#1949C2214yescomplemented
57#1947C2214yescomplemented
58#1921C2214yescomplemented
59#1919C2214yescomplemented
60#1918C2214yescomplemented
61#1918C2214yescomplemented
62#1854C2214yes—
63#1853C2212yes—
64#1852C2212yes—
65#1852C2212yes—
66#1852C2212yes—
67#1852C2214yes—
68#1851C2212yes—
69#1851C2212yes—
70#1850C2214yes—
71#1825C2212yes—
72#1825C2212yes—
73#1825C2214yes—
74#1824C2211yescharacteristic
75#1824C2212yes—
76#1824C2211yescharacteristic
77#1823C2211yescharacteristic
78#1823C2212yes—
79#1823C2211yescharacteristic
80#1809C2214yes—
81#1808C2212yes—
82#1808C2212yes—
83#1797C2212yes—
84#1797C2212yes—
85#1736C2214yes—
86#1735C2214yes—
87C24×C4C2311yescharacteristic
88C22×M16C2312yes—
89C23×C8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#125C2312yes—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117—24no—
97C2×#117Q818yescomplemented
98C2×#117C2312yes—
99C2×#117C2312yes—
100C2×#117C2312yes—
101C22×C4⋊C4—22no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4D412yes—
110C22×C42—21no—
111C22×C42—22no—
112C22×C42C2312yes—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×C4⋊C8C2314yes—
119#407—24no—
120C2×C22⋊C8—24no—
121C2×C22⋊C8C2311yescharacteristic
122C2×C22⋊C8C2311yescharacteristic
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126#400—22no—
127#399—22no—
128#398—22no—
129#398—22no—
130#396—22no—
131#396—22no—
132#395—22no—
133#393—22no—
134#391—22no—
135#391—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389C2314yes—
141#389C2314yes—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#389C2312yes—
146#388—24no—
147#388—24no—
148#388—24no—
149#388—24no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#384C2312yes—
159#383—24no—
160#383—22no—
161#383—22no—
162#382C2312yes—
163#381—21no—
164#380—22no—
165#380—21no—
166#380—21no—
167#380—21no—
168#380—21no—
169#380—22no—
170#380C2311yescharacteristic
171#380C2311yescharacteristic
172#380C2312yes—
173#380C2312yes—
174#380C2312yes—
175#380C2311yescharacteristic
176#380C2312yes—
177#380C2311yescharacteristic
178#379—21no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—24no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4D412yes—
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C4×C22⋊C4C2312yes—
192C4×C22⋊C4C2311yescharacteristic
193C4×C22⋊C4C2311yescharacteristic
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—22no—
197C2×#95—22no—
198C2×#95D411yescharacteristic
199C2×#95D411yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25C2×D411yescharacteristic
215C22×Q8—22no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—44no—
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×Q814yes—
227C23×C4#4114yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C2×M16—24no—
232C2×M16—44no—
233C22×C8—24no—
234C22×C8—44no—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2414yes—
238C4⋊Q8—44no—
239#125—44no—
240C22⋊Q8—48no—
241C22⋊Q8—84no—
242#117—28no—
243#117—28no—
244#117—24no—
245#117—24no—
246#117—44no—
247#117C2×Q818yes—
248#117C2×Q818yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4C2×Q814yes—
270C2×C4⋊C4C2×Q814yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—82no—
292C2×C22⋊C4—82no—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D412yes—
297C2×C42—28no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—44no—
304C2×C42—84no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2412yes—
308C2×C42C2412yes—
309#107—44no—
310#107—44no—
311#107—44no—
312#106—44no—
313C4⋊C8—44no—
314C4⋊C8—44no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—21no—
325#95—22no—
326#95—21no—
327#95—24no—
328#95—24no—
329#95—21no—
330#95—21no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95C2×D412yes—
339#95C2×D411yescharacteristic
340#95C2×D411yescharacteristic
341#95C2×D412yes—
342#95C2×D411yescharacteristic
343#95C2×D411yescharacteristic
344C24—22no—
345C24—22no—
346C24—42no—
347C24—42no—
348C24C22⋊Q814yes—
349C24C22×D411yescharacteristic
350C24C22×D411yescharacteristic
351C24C22×D411yescharacteristic
352C2×Q8—44no—
353C2×Q8—82no—
354C2×Q8—82no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4C22⋊Q814yes—
388C22×C4C22⋊Q814yes—
389C22×C4C22⋊Q814yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×Q812yes—
398C22×C4C22×Q812yes—
399C22×C4C22×Q812yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2511yescharacteristic
404M16—84no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—84no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4C22×Q814yes—
437C4⋊C4C22×Q814yes—
438C22⋊C4—28no—
439C22⋊C4—28no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—81no—
451C22⋊C4—81no—
452C22⋊C4—82no—
453C42—28no—
454C42—28no—
455C42—24no—
456C42—24no—
457C42—42no—
458C42—42no—
459C42—84no—
460C42—84no—
461C42—82no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—28no—
473C23—44no—
474C23—44no—
475C23—84no—
476C23—84no—
477C23C2×C22⋊Q812yes—
478C23C2×C22⋊Q812yes—
479C23C2×C22⋊Q812yes—
480C23#57612yes—
481C23#57812yes—
482C23C23×D411yescharacteristic
483Q8—164no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—84no—
531C2×C4—88no—
532C2×C4—164no—
533C2×C4C2×C22⋊Q812yes—
534C2×C4C2×C22⋊Q812yes—
535C2×C4C2×C22⋊Q812yes—
536C2×C4#57514yes—
537C2×C4C23×Q811yescharacteristic
538C2×C4C22×#4111yescharacteristic
539C2×C4#58514yes—
540C8—42no—
541C8—42no—
542C8—44no—
543C22—22no—
544C22—24no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—88no—
558C22#308314yes—
559C22C22×C22⋊Q811yescharacteristic
560C22#348512yes—
561C22C2×#57512yes—
562C22C2×#57611yescharacteristic
563C22C2×#57811yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—24no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—44no—
572C4—44no—
573C4—84no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#345812yes—
579C4#349112yes—
580C2—24no—
581C2—42no—
582C2—42no—
583C2#3509011yescharacteristic
584C2#6048411yescharacteristic
585C2C2×#308311yescharacteristic
586C2#6172712yes—
587C2#6173012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33151C2semidirect
4. #33146C2semidirect
5. #29973C2semidirect
6. #29969C2semidirect
7. #26015C2semidirect
8. #26005C2semidirect
9. #25997C2semidirect
10. #25989C2semidirect
46. #2101C22semidirect
47. #2100C22semidirect
48. #2088C22semidirect
49. #2087C22semidirect
50. #2087C22semidirect
51. #2086C22semidirect
52. #2059C22semidirect
53. #2057C22semidirect
54. #1989C22semidirect
55. #1988C22semidirect
56. #1949C22semidirect
57. #1947C22semidirect
58. #1921C22semidirect
59. #1919C22semidirect
60. #1918C22semidirect
61. #1918C22semidirect
97. C2×#117Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#350792
C2×#23472
#331512
#331462
#299732
#299692
#260152
#260052
#259972
#259892
#219442
#219402
#219322
#207022
#206992